A355291 Expansion of e.g.f. exp(exp(x)*(exp(x) + 1) - 2).

1, 3, 14, 81, 551, 4266, 36803, 348543, 3583484, 39652659, 468970211, 5894584812, 78366374813, 1097537989671, 16136598952718, 248309032411485, 3988468487017379, 66715970326561170, 1159712730763363991, 20909709414253764819, 390374806223071148084, 7534929383736826736007

Offset: 0

Author

Vaclav Kotesovec, Jun 27 2022

Keywords

• nonn

Comments

In general, if m > 0, b > d >= 1 and e.g.f. = exp(m*exp(b*x) + r*exp(d*x) + s) then a(n) ~ exp(m*exp(b*z) + r*exp(d*z) + s - n) * (n/z)^(n + 1/2) / sqrt(m*b*(1 + b*z)*exp(b*z) + r*d*(1 + d*z)*exp(d*z)), where z = LambertW(n/m)/b - 1/(d + b/LambertW(n/m) + b^2 * m^(d/b) * n^(1 - d/b) * (1 + LambertW(n/m)) / (d*r*LambertW(n/m)^(2 - d/b))). - Vaclav Kotesovec, Jul 03 2022

In addition, if b/d >=2 then a(n) ~ c * (b*n/LambertW(n/m))^n * exp(n/LambertW(n/m) + r * (n/(m*LambertW(n/m)))^(d/b) - n + s) / sqrt(1 + LambertW(n/m)), where c = 1 for b/d > 2 and c = exp(-r^2/(8*m)) for b/d = 2. - Vaclav Kotesovec, Jul 10 2022

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.

Crossrefs

Cf. A001861, A055882, A126390, A143405, A355379.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[Exp[2*x] - 2 + Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x)*(exp(x) + 1) - 2))) \\ Michel Marcus, Jun 27 2022

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * Bell(k) * Bell(n-k).

a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jul 01 2022

a(n) ~ exp(exp(2*z) + exp(z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(2*(1 + 2*z)*exp(2*z) + (1 + z)*exp(z)), where z = LambertW(n)/2 - 1/(1 + 2/LambertW(n) + 4 * n^(1/2) * (1 + LambertW(n)) / LambertW(n)^(3/2)). - Vaclav Kotesovec, Jul 03 2022

a(n) ~ 2^n * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n * exp(n + 17/8 - n/LambertW(n) - sqrt(n/LambertW(n)))). - Vaclav Kotesovec, Jul 08 2022